Sum of Soft Topological Spaces

Tareq M Al-Shami,Baravan A Asaad,Ljubiša D R Kočinac

doi:10.3390/math8060990

Tareq M Al-Shami, Baravan A Asaad + Show 1 more

Open Access

https://doi.org/10.3390/math8060990

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Abstract

In this paper, we introduce the concept of sum of soft topological spaces using pairwise disjoint soft topological spaces and study its basic properties. Then, we define additive and finitely additive properties which are considered a link between soft topological spaces and their sum. In this regard, we show that the properties of being p-soft T i , soft paracompactness, soft extremally disconnectedness, and soft continuity are additive. We provide some examples to elucidate that soft compactness and soft separability are finitely additive; however, soft hyperconnected, soft indiscrete, and door soft spaces are not finitely additive. In addition, we prove that soft interior, soft closure, soft limit, and soft boundary points are interchangeable between soft topological spaces and their sum. This helps to obtain some results related to some important generalized soft open sets. Finally, we observe under which conditions a soft topological space represents the sum of some soft topological spaces.

Highlights

In 1999, Molodtsov [1] introduced the concept of soft sets as an innovative approach to deal with uncertainties
Our results mainly investigate invariant properties between soft topological spaces and their sum
In the following two examples, we show that the properties of indiscrete and door soft spaces are not additive properties

Summary

Introduction

In 1999, Molodtsov [1] introduced the concept of soft sets as an innovative approach to deal with uncertainties. Ali et al [4] have made some amendments for some results obtained by [3] and have defined new types of soft union and intersection between two soft sets. The study of generalized soft open sets began by Chen [25] He defined soft semi-open sets and discussed main properties. In [31], the authors have presented the concept of the pointwise topology of soft topological spaces and investigated the properties of soft mapping spaces. We aim through this work to introduce and study the concept of sum of soft topological spaces using pairwise disjoint soft topological spaces. We made use of interchangeability of soft interior and soft closure operators between soft topological spaces and their sum to obtain some results related to some important generalized soft open sets. We study under what conditions a soft topological space represents the sum of some soft topological spaces

Soft Sets

Soft Topology

Sum of Soft Topological Spaces

Conclusions